Revisiting the Linear Model of Coregionalization

Abstract

The linear coregionalization model (LCM) is the model most commonly used in practice for cokriging. The LCM model amounts to represent the observed variables as linear combinations of sets of independent (or at least uncorrelated) underlying variables. The popularity of the LCM stems mostly from the ease of modeling and verification of the admissibility of the model. However, LCM has strong limitations as it implies symmetrical cross-covariances for the variables under study. This symmetry could explain why the cokriging is generally unsuccessful to improve significantly over kriging when the observed variables are collocated. One possible generalization to the LCM (GLCM) is to consider the observed variables as linear combinations of underlying independent variables and of variables representing deterministic functions of some of these underlying variables. Candidate functions are for example regularization, spatial shift and derivatives, the last two functions enabling to introduce asymmetry (and anisotropy) in the cross-covariances when needed. We note that GLCM comprises the deterministic cokriging of a variable and a given function of it (partial derivative, regularization, …) as a particular case. To ensure admissibility of the model, the modeling is done directly on the coefficients of the underlying variables, and functions of it, using an iterative non-linear optimization approach that compares the theoretical covariances computed from the GLCM to the experimental ones. Various synthetic cases are presented illustrating the flexibility of the GLCM. The well-known Gslib data file is also used to compare the relative performances of the LCM and of the GLCM. GLCM is shown to provide more precise estimations than the LCM model for the test case.